Matrix-free stability analysis of fluid-structure interaction problems with an Immersed Boundary method.

The mutual interaction between viscous fluids and elastic solids plays a role in a variety of engineering applications and natural systems. In some cases, large structural deformations are sought for engineering purposes, at other times undesirable vibrations cause structural damage. Performing accurate and high-fidelity numerical simulations to investigate the behavior of such fluid-structure interaction (FSI) problems is not trivial nor cheap, thus the development of numerical methods for FSI is a very active research area. Another possibility is to investigate the linear dynamics of these coupled systems. Linear stability analysis is a popular tool in the fluid mechanics community since it allows the faster identification of the stability criteria and gives insights into the underlying physics of the phenomena. As a matter of fact, the data resulting from temporal simulations may be difficult to interpret directly. Besides, it is usually necessary to run long simulations to go past the transient phase and see if a given perturbation fades away. However, many FSI configurations of engineering relevance are still uninvestigated from a linear point of view due to the complexity of the linearization of the coupled dynamics. The main objective of this thesis is to promote a wide adoption of the linear approach to FSI problems. An Immersed Boundary (IB) framework is introduced, based on a direct-forcing moving-least-squares procedure to couple the fluid and solid dynamics, which has been already well validated and has proven to accurately capture the coupled dynamics. The major novelty of the present work is the development of a general approach to perform linear stability analyses of large-scale FSI problems, based on the IB method previously mentioned. To the author’s knowledge, in the context of FSI systems, the global linear approach has not been yet extended to problems involving multiple elastic bodies. The proposed methodology allows the treatment of multi-body configurations with no added complexity and reasonable computational cost. In this thesis, the proposed methodology is derived and the numerical solver is validated against results from the literature. Then, the procedure is applied to analyze the vortex-induced vibrations of two elastically mounted cylinders in tandem arrangement. Two unstable eigenmodes are identified in the analysis, and an explanation is suggested for a change in the nonlinear behavior of the system, previously noted by other researchers but still without interpretation. In the last chapter, a different methodology is adopted to investigate the linear dynamics of a gas bubble placed in a uniform straining flow. For this case, it is used a recently developed linearized Arbitrary Lagrangian-Eulerian framework. The linear analysis of this configuration reveals the existence of a saddle-node bifurcation linked to the breakup of the bubble via an end-pinching mechanism. Interestingly, a self-propelling unstable mode emerges, which is counterintuitive as it consists in a displacement of the bubble towards a higher-pressure region. The existence of this mode is confirmed in the inviscid limit, and it is shown that the propulsive mechanism exploits shape asymmetries to create a net thrust.